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Theorem xrmaxleim 10852
Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
Assertion
Ref Expression
xrmaxleim  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )

Proof of Theorem xrmaxleim
Dummy variables  f  g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrlttri3 9424 . . . 4  |-  ( ( f  e.  RR*  /\  g  e.  RR* )  ->  (
f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
21adantl 273 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  (
f  e.  RR*  /\  g  e.  RR* ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 simplr 500 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  B  e.  RR* )
4 prid2g 3575 . . . 4  |-  ( B  e.  RR*  ->  B  e. 
{ A ,  B } )
53, 4syl 14 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  B  e.  { A ,  B }
)
6 simpllr 504 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  <_  B )
7 xrlenlt 7701 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  <->  -.  B  <  A ) )
87ad3antrrr 479 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( A  <_  B  <->  -.  B  <  A ) )
96, 8mpbid 146 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  A )
10 breq2 3879 . . . . . . 7  |-  ( y  =  A  ->  ( B  <  y  <->  B  <  A ) )
1110notbid 633 . . . . . 6  |-  ( y  =  A  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1211adantl 273 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
139, 12mpbird 166 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  y )
14 xrltnr 9407 . . . . . 6  |-  ( B  e.  RR*  ->  -.  B  <  B )
1514ad4antlr 482 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  B )
16 breq2 3879 . . . . . . 7  |-  ( y  =  B  ->  ( B  <  y  <->  B  <  B ) )
1716notbid 633 . . . . . 6  |-  ( y  =  B  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1817adantl 273 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1915, 18mpbird 166 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  y )
20 elpri 3497 . . . . 5  |-  ( y  e.  { A ,  B }  ->  ( y  =  A  \/  y  =  B ) )
2120adantl 273 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  (
y  =  A  \/  y  =  B )
)
2213, 19, 21mpjaodan 753 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  -.  B  <  y )
232, 3, 5, 22supmaxti 6806 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  A  <_  B )  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B )
2423ex 114 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR* ,  <  )  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670    = wceq 1299    e. wcel 1448   {cpr 3475   class class class wbr 3875   supcsup 6784   RR*cxr 7671    < clt 7672    <_ cle 7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-ltirr 7607  ax-pre-apti 7610
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-iota 5024  df-riota 5662  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678
This theorem is referenced by:  xrmaxltsup  10866  xrmaxadd  10869  xrmineqinf  10877
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